If the system of linear equations. 2x + 2ay + az = 0 2x + 3by + bz = 0 2x + 4cy + cz = 0, where a, b, c ∈ R are non-zero and distinct; has a non-zero solution, then

Engineering

Mathematics

Solving System of Linear Equation Cramers Rule

Question

If the system of linear equations.

2x + 2ay + az = 0

2x + 3by + bz = 0

2x + 4cy + cz = 0,

where a, b, c ∈ R are non-zero and distinct; has a non-zero solution, then

a + b + c = 0

a, b, c are in A.P.

$\frac{1}{a}, \frac{1}{b}, \frac{1}{c}$ are in A.P.

a, b, c are in G.P.

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For non-trivial solution

$| \begin{matrix} 2 & 2 a & a \\ 2 & 3 b & b \\ 2 & 4 c & c \end{matrix} | = 0$

$| \begin{matrix} 1 & 2 a & a \\ 1 & 3 b & b \\ 1 & 4 c & c \end{matrix} | = 0$

(3bc – 4bc) – (2ac – 4ac) + (2ab – 3ab) = 0

–bc + 2ac – ab = 0

ab + bc = 2ac

a, b, c in H.P.

⇒ $\frac{1}{a}, \frac{1}{b}, \frac{1}{c} in A . P .$